Photon wave mechanics and position eigenvectors

Abstract

One and two photon wave functions are derived by projecting the quantum state vector onto simultaneous eigenvectors of the number operator and a recently constructed photon position operator [Phys. Rev A 59, 954 (1999)] that couples spin and orbital angular momentum. While only the Landau-Peierls wave function defines a positive definite photon density, a similarity transformation to a biorthogonal field-potential pair of positive frequency solutions of Maxwell's equations preserves eigenvalues and expectation values. We show that this real space description of photons is compatible with all of the usual rules of quantum mechanics and provides a framework for understanding the relationships amongst different forms of the photon wave function in the literature. It also gives a quantum picture of the optical angular momentum of beams that applies to both one photon and coherent states. According to the rules of qunatum mechanics, this wave function gives the probability to count a photon at any position in space.

A new look at the position operator in quantum theory

“…The author of Ref. [35] discusses generalizations of the photon position operator proposed by Pryce [36]. However, the Pryce operator and its generalizations discussed in Refs.…”

“…The dependence of the momentum wave function on t is now defined by Eq. (35). In full analogy with the derivation of Eq.…”

“…However, the Pryce operator and its generalizations discussed in Refs. [34,35] differ from the Newton-Wigner operator only by terms of the order of the wave length. Hence in semiclassical approximation all those operators are equivalent.…”

“…Ref. [35]). In this approach the photon is described by a local field and the momentum and coordinate representations are related to each other by standard Fourier transform.…”

Photon Localization Revisited

“…However, just before the turn of the century both of these widely-espoused notions were disproved (Bialynicki-Birula, 1998;Hawton, 1999) and in the new century a fresh interest in the photon localization problem seems to have been awakened, see, e.g., (Bialynicki-Birula & Bialynicka-Birula, 2009;Chan et al, 2002;Hawton, 2007;Keller, 2000;2005), meeting the needs of developments in near-field optics, cavity QED, and quantum computing. Recently, into the study of quantum phenomena in general and photon localization in particular, the so-called localized waves were involved (Belgiorno et al, 2010;Besieris et al, 1994;Ciattoni & Conti, 2007;Jáuregui & Hacyan, 2005;Saari et al, 2005).…”